One night in London, even a night at the O2, is small beer compared to Taylor Swift's record-breaking success back home in the US. There, her recent single Mine topped the iTunes chart within six hours; here, it limped to No 30. US critics praise the 21-year-old's songwriting chops – old-fashioned Nashville craft, through the prism of a high-school movie – while their British equivalents find the sheer efficiency of her slick, catchy pop-rock hard to get excited about.

This isn't a problem for the O2 fans, whose age and gender are advertised by the shortness of the bar queue and the pitch of the screaming, which is of an intensity usually inspired by boy-band torsos. Parents reluctantly chaperoning their daughters could do a lot worse. Appearing in a tasselled gold dress and high leather boots, Swift is like a cool, talented big sister: approachably attractive, warmly solicitous, modestly self-assured.

Faced with gale-force shrieking between songs, she pulls a "little old me?" face, like the pretty but geeky girl who is unexpectedly voted prom queen. Capable but unspectacular as a singer, guitarist and pianist (the backing band do the heavy lifting), Swift is a prodigiously confident songwriter. All her songs, she admits, are "about love and horrible break-ups", but it's the way she tells them. Mine pivots on the elegant line: "You made a rebel of a careless man's careful daughter."

Fifteen, which she performs solo from a smaller stage at the back of the arena, is a pitch-perfect, detail-rich portrait of adolescent longing. Speak Now manages to make an account of ruining someone's wedding not just funny but sweet. For the hefty percentage of the audience too young to have experienced real passion or heartbreak, Swift's songs are like a potential road map. Her lyrics, like her peppy between-song chats, promise things will work out OK.

The 90-minute show falls prey to some arena-rock stand-bys such as the hacky, soft-rock guitar solo that buys time for a costume change and the self-aggrandising video montage of award show triumphs, in which the "little old me?" face makes several appearances. But like the most instinctive pop stars and politicians, Swift has the ability to flirt with 20,000 people at once, and so likably that you can forgive the occasional bit of shtick.

"Would it be incredibly nerdy to ask if I could get a picture of you all on my cellphone?" she asks the screaming throng. As if.